Triangles with non rational sides and integral area

Consider the triangle with sides √5, √65 and √68. It can be shown that this triangle has area 9.

S(n) is the sum of the areas of all triangles with sides √(1+b²), √(1+c²) and √(b²+c²) (for positive integers b and c ) that have an integral area not exceeding n.

The example triangle has b=2 and c=8.

S(10⁶)=18018206.

Find S(10¹⁰).

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